At first, we consider the parabola, the lines and.
Again, we use the Graphmatica software to obtain four intersection points, repeat from step a to step c. Then label the intersections on the graph shown below.
The x-values of these intersections from the left to the right on the x-axis:
-
x1≈ 1.807
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x2≈ 2.268
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x3≈ 5.732
-
x4≈ 7.193
Find the values of and and name them respectively SL and SR.
SL = x2−x1≈ 2.268−1.807≈ 0.461
SR = x4−x3≈ 7.193-5.732≈ 1.461
Finally, calculate.
Secondly, we consider the parabola, the lines and.
By using the manual calculation, we can calculate the four intersections.
Calculate the intersections between f(x) and h(x):
(1)
(2)
Substitute (2) into (1):
or .
Using calculator to obtain the approximate values of x
or
(x, y) = (3.823; 7.646)
(x, y) = (1.177; 2.354)
Calculate the intersections between f(x) and g(x):
(3)
(4)
Substitute (4) into (3):
or
(x, y) = (3; 3)
(x, y) = (1.5; 1.5)
Hence, the x-values from left to right are:
-
x1≈ 1.177
-
x2= 1.500
-
x3= 3.000
-
x4≈ 3.823
Calculation of SL and SR:
SL = x2−x1≈ 1.500−1.177≈ 0.323
SR = x4−x3≈ 3.823−3.000≈ 0.823
Calculate.
Thirdly, consider parabola, the lines and.
Using the same method, we apply the Graphmatica software to determine the four intersections which are illustrated below:
The x-values of these intersections from the left to the right on the x-axis:
-
x1≈ 0.726
-
x2= 1.000
-
x3≈ 1.167
-
x4≈ 1.608
Calculation of SL and SR:
SL = x2−x1≈ 1.000− 0.726≈ 0.274
SR = x4−x3≈ 1.608−1.167≈ 0.441
Calculate.
Next, consider parabola , the lines and.
By using Graphmatica software, we can obtain the four intersections of parabola , the lines and.
The x-values of these intersections from the left to the right on the x-axis:
-
x1≈ 2.417
-
x2≈ 3.172
-
x3≈ 8.828
-
x4≈ 11.583
Calculation of SL and SR:
SL = x2−x1≈ 3.172− 2.417≈ 0.755
SR = x4−x3≈ 11.583−8.828≈ 2.755
Calculate.
Consider parabola, the lines and.
By using Graphmatica software, we can obtain the four intersections of parabola , the lines and.
The x-values of these intersections from the left to the right on the x-axis:
-
x1≈ 2.591
-
x2≈ 3.285
-
x3≈ 9.315
-
x4≈ 11.809
Calculation of SL and SR:
SL = x2−x1≈ 3.285− 2.591≈ 0.694
SR = x4−x3≈ 11.809−9.315≈ 2.494
Calculate.
Table shows the values of D for parabolas of the form, with vertices in quadrant 1, intersected by the lines and.
As can be seen from the table D is inversely proportional to the value of a.
The values of D for parabolas of the form, with vertices in quadrant 1, intersected by the lines and, are inversely proportional to the values of a.
3. Investigate your conjecture for any real value of a and any placement of the vertex. Maintain the labeling convention used in parts 1 and 2 by having the intersections of the first line to be x2 and x3 and the intersections with the second line to be x1 and x4.
Consider parabola , the lines and.
Determine the four intersections which are illustrated below:
The x-values of these intersections from the left to the right on the x-axis:
-
x1≈ −5.236
-
x2≈ −4.000
-
x3≈ −1.000
-
x4≈ −0.764
Calculation of SL and SR:
SL = x2−x1≈ −4.000− (−5.236) ≈ 1.236
SR = x4−x3≈ −0.794− (−1.000) ≈ 0.206
Calculate.
The conjecture does not hold true because due to part 2, so in this case when a = -1, D should be:.
Therefore, the conjecture should be,.
Consider the parabola , the lines and.
The four intersections between the parabola , the lines and can be found by again using Graphmatica software.
The x-values of these intersections from the left to the right on the x-axis:
-
x1≈ 1.149
-
x2≈ 1.382
-
x3≈ 3.618
-
x4≈ 4.351
Calculation of SL and SR:
SL = x2−x1≈ 1.382− 1.149≈ 0.233
SR = x4−x3≈ 4.351−3.618≈ 0.733
Calculate.
Consider the parabola , the lines and.
By using Graphmatica software, we can obtain the four intersections of parabola , the lines and.
The x-values of these intersections from the left to the right on the x-axis:
-
x1≈ 0.451
-
x2≈ 0.500
-
x3≈ 5.000
-
x4≈ 5.550
Calculation of SL and SR:
SL = x2−x1≈ 0.500− 0.451≈ 0.049
SR = x4−x3≈ 5.550−5.000≈ 0.550
Calculate.
≈ 0.5
The results of investigating different real values of a and placement of the vertex :
Proof:
Find the two intersections between parabola f(x) and g(x):
or
Find the two intersections between parabola f(x) and h(x):
or
Hence, the conjecture about the values of D, for all real values of a, .
4. Investigating the conjecture when the intersecting lines are changed.
We will still use the same parabola but the intersecting lines will be varied
The general equation of intersecting lines is
As there are two intersecting lines, they should be written as the following equations:
Consider parabola and the lines and.
The intersection points then can be found by using Graphmatica software.
-
x1≈ 1.258
-
x2≈ 1.459
-
x3≈ 7.541
-
x4≈ 8.742
Calculation of SL and SR:
SL = x2−x1≈ 1.459− 1.258≈ 0.201
SR = x4−x3≈ 8.742−7.541≈ 1.201
Calculate.
In this case, the conjecture holds true.
Consider parabola and the lines and
.
By using Graphmatica software, we can obtain the four intersections of parabola , the lines and.
-
x1≈ 0
-
x2≈ 0.392
-
x3≈ 4.000
-
x4≈ 5.109
Calculation of SL and SR:
SL = x2−x1≈ 0.392− 0≈ 0.392
SR = x4−x3≈ 5.109−4.000≈ 1.109
Calculate.
Consider parabola and the lines and
Again, using Graphmatica software, we can obtain the four intersection points.
-
x1≈ -2.000
-
x2≈ 0.209
-
x3≈ 1.500
-
x4≈ 4.791
Calculation of SL and SR:
SL = x2−x1≈ 0.209+2.000≈ 2.209
SR = x4−x3≈ 4.791−1.500≈ 3.291
Calculate.
Consider parabola and the line y = 5 and y =-3.
By using Graphmatica software, we can obtain the four intersection points.
-
x1≈ -6.123
-
x2≈ -5.000
-
x3≈ 1.000
-
x4≈ 2.123
Calculation of SL and SR:
SL = x2−x1≈ (-5.000) − (-6.123) ≈ 1.123
SR = x4−x3≈ 2.123−1.000≈ 1.123
Calculate.
Proof:
ax2 + (b− m1) x + c= 0
ax2 + (b− m2) x + c= 0
D=|SL−SR| = |(x2−x1) – (x4−x3)|= |x2−x1− x4+x3|
= |x2+x3−x1− x4|= |(x2+x3) −(x1+x4)|
x2 +x3 = 2[−(b− m1)/2a]= −(b− m1)/a
x1 +x4 = 2[−(b− m2)/2a]= −(b− m2)/a
, with a R.
